Definitions from universal algebra

Definition in terms of linear representations

A subgroup of a group is termed normal in if and only if there exists a linear representation of over a field of characteristic zero, with the property that the character of the representation is nonzero on all elements of , and zero on all elements outside .

A subgroup of a group is termed normal in if and only if the trivial linear representation of over characteristic zero, induces a representation of (by induction of representations) that is zero on all elements outside .

Definitions in terms of generating sets

A subgroup of a group is normal in if, whenever is a generating set of and is a generating set of , and for all . (Note that this is the definition used to test normality -- Further information: Normality testing problem

Suppose is a group and is a subgroup. Suppose we quotient out by relations of the form for all . Then is normal if and only if the only elements that become trivial in the quotient, are those that originally came from .